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Properties

Label 2-450-45.32-c1-0-15
Degree $2$
Conductor $450$
Sign $0.886 + 0.461i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (1.22 − 1.22i)3-s + (0.866 + 0.499i)4-s + (1.49 − 0.866i)6-s + (0.707 + 0.707i)8-s − 2.99i·9-s + (3 − 1.73i)11-s + (1.67 − 0.448i)12-s + (−0.896 − 3.34i)13-s + (0.500 + 0.866i)16-s + (−4.24 + 4.24i)17-s + (0.776 − 2.89i)18-s + 5i·19-s + (3.34 − 0.896i)22-s + (5.79 − 1.55i)23-s + 1.73·24-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.707 − 0.707i)3-s + (0.433 + 0.249i)4-s + (0.612 − 0.353i)6-s + (0.249 + 0.249i)8-s − 0.999i·9-s + (0.904 − 0.522i)11-s + (0.482 − 0.129i)12-s + (−0.248 − 0.928i)13-s + (0.125 + 0.216i)16-s + (−1.02 + 1.02i)17-s + (0.183 − 0.683i)18-s + 1.14i·19-s + (0.713 − 0.191i)22-s + (1.20 − 0.323i)23-s + 0.353·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.886 + 0.461i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.886 + 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.51749 - 0.616353i\)
\(L(\frac12)\) \(\approx\) \(2.51749 - 0.616353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.896 + 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
19 \( 1 - 5iT - 19T^{2} \)
23 \( 1 + (-5.79 + 1.55i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.46 - 6i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (11.7 + 3.13i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.36 + 2.24i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (8.57 - 8.57i)T - 73iT^{2} \)
79 \( 1 + (12.1 - 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.32 - 8.69i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (1.34 - 5.01i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.16907115118549649471625938454, −10.22869320602536941912092158691, −8.800983183283094892270454633483, −8.376376760814373202120032102337, −7.12808640207112292753245823761, −6.48326488800916684402510970762, −5.39955836452924770817693692195, −3.91651305770200005772085446846, −3.06321632581576803666626319939, −1.57464320993199075298202035406, 2.06128004375683071793739145504, 3.19970530146079623667454429320, 4.46764115554667495524249801523, 4.89613403437103222119380555993, 6.58904792784930505759929629100, 7.27016137855576399949468147700, 8.760477454575250402687952624962, 9.353279558175682822960407410835, 10.19766721864716910446485180988, 11.47625788998113095445129051805

Graph of the $Z$-function along the critical line